ZDM 2001 Vol. 33 (3)
Baumslag, Benjamin:
Fundamentals of Teaching
Mathematics at University Level
London: Imperial College Press, 2000. – 240p.
ISBN 1-86094-214-8
Ed Dubinsky, New York (USA)
This is a book of opinions. The author has taught
mathematics at the university level for many in years, in
several different universities in a number of countries. He
has done this in a conscientious manner, attempting to be
aware of everything that has been going on and trying to
do the best job of teaching that he could. As a result, he
has developed points of view on a vast number of topics,
issues, ideas, etc. related to university level mathematics
teaching and the purpose of this book is to share those
opinions with readers. The material is quite
comprehensive and just about every aspect, from how a
teacher should stand in relation to the blackboard to
research in learning mathematics is treated. Although the
book might be useful to someone who had not the
slightest idea about the enterprise of teaching university
mathematics, I did not find any of the opinions to be
remarkable or surprising. The most superficial topics
(such as checklists for how to discuss the statement of a
theorem, or what kinds of breaks to take in a lecture) are
given detailed treatment whereas major issues of teaching
and learning (such as those related to the mathematical
concepts of function or limit) are handled in analmost
offhand manner.
For example, the author spends two pages giving three
methods for teaching limits. The first is for the teacher to
introduce the idea of successive approximations, the
second is for the teacher to describe the limit informally
as the expected value based on neighboring information,
and the third is for the teacher to describe the epsilondelta idea in terms of controlling error. It is certainly
useful for beginning faculty to know that such methods
are used by experienced teachers to help students
understand deep and difficult concepts. But it would be
perhaps even more useful for the neophyte to be aware
that, in spite of such methods, very few undergraduate
mathematics students develop very much understanding
of such concepts.
We see in this example, two major characteristics of
Baumslag’s point of view. One is the emphasis on what
he (and others) refer to as “teacher centred” teaching.
Although many other approaches are mentioned, it is
clear throughout the book that Baumslag’s emphasis is on
what the teacher does: present, describe, introduce, and so
on. There is very little about what may or may not be
happening with the students – and this relates to the
second major characteristic of this work. Again and again
an opinion is stated about what is a good or a bad thing
for the teacher to do. This is, however, almost never
related to student learning. That is, Baumslag makes not
the slightest attempt to justify any of his many opinions
on the basis of student learning. He does of course say
Book Reviews
that this or the other suggestion for what a teacher might
say or do will lead to student learning, but he gives us no
way of making anything like an independent judgement.
He tells us about his many years of experience (but not of
any exceptional successes) and asks us to more or less
take his word for it.
Indeed, Baumslag seems almost defensive about
suggestions that alternative approaches to teaching can
lead to results that are better than what happens in
traditional teaching. Consider, for example, the following
curious excerpt.
“One of my colleagues explained with great pride that the new
method of teaching he had introducedhad meant that the
students had been forced to work much harder. As a
consequence, they had learneda lot more. He therefore felt that
his new method of teaching was a success. I cannot agree. I find
itquite natural that if somebody works harder in an effective
manner he will learn much more.” (p. 49.)
In my view, the key here is the term “effective”. There
are many university situations in which it is not difficult
to get students to work very hard. I have conducted
courses in mathematics to engineers at several
universities, some of them amongst the highest ranked. In
these experiences, I observed that students worked
extremely hard, but they did not learn very much
mathematics (beyond the ability to repeat precisely any
calculations that had been carefully, explicitly, and fully
laid out for them). It was only when teaching methods
could be found that got them to think for themselves
about mathematical problem situations, and still spend
many hours working at mathematics, that exceptional
learning took place. So if we can take at face value (since
Baumslag does not deny it) that his colleague’s students
worked harder than normal and learned a lot more, then I
think it is fair to agree with the conclusion that the
method was a success. Indeed, I and most of the
mathematics faculty I know are overjoyed when they
have such results – no matter what the method of
teaching!
Although we may well concede that Baumslag is
entitled to his opinions, I must say that I am rather
disturbed by the statement he makes in the very next
sentence after the above quote.
“The whole point of providing instruction is to reduce the work
required for learning.”
I taught college level mathematics for 44 years and I find
one conclusion unavoidable. Learning mathematics is
hard work and learning more mathematics is more hard
work. Baumslag goes on to say that he would have been
impressed if his colleague had claimed that the students
had “learnt more with less work.” I worry about teachers
whose goal is to get students to do less work. It is easy to
create a false impression of successful learning, for
example by giving exams with questions that appear hard
but are relatively easy for the students because they have
done little else but drill on prototypes of precisely those
questions. Your students will love you, the administration
will be happy, and you may win teaching awards. But
your students will not learn.
These are not the only opinions Baumslag puts forth
that I find unfortunate. Here is a sampling of some others,
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followed in some cases by opinions of my own..
“When I was at school, I studied Euclidean Geometry, from the
age of 12. It was a matter of proving theorems from axioms and
definitions. I and my contemporaries had a very good idea of
what a proof entailed.” (p. 19.)
This high ability in mathematics on the part of students in
the past is a belief held by so many people that no one
seems to think it needs to be supported by any facts. The
only data I know of suggests the opposite: students in
mathematics throughout the 20th Century were, on
average, not substantially better at mathematics than
those of today.
“The event of the computer means that people can now solve
extremely difficult problems with less knowledge than before.”
(p. 25.)
“Explaining and giving drill on using the rule is training,
explaining why the rule works is education.” (p. 31.)
At the risk of repeating myself, I must counter this
opinion with the view that explanations are never
education. Getting the students to understand why the
rule works is education.
“Not everybody can learn mathematics.” (p. 33.)
In a general consideration of new methods of teaching,
Baumslag writes:
“The problem is reminiscent of the alchemists search for
converting lead into gold; nobody knew whether it was possible
or economic, and much time and energy were frittered away.”
(p. 49.)
It is particularly disturbing to think of so many years of
teaching to so many students has been conducted by one
who has such a view about efforts to explore the
unknown in relation to teaching and learning. I have
recently read that Isaac Newton was very interested,
perhaps in an active way, in alchemy. If he were living in
Newton’s time, would Baumslag have said that Newton
was frittering his time away when he was trying to
understand the relation between lead and gold but not
when he was trying to understand the influence, at a
distance, each could have on the other’s motion?
“Teacher-centred teaching has always been effective and is one
of the most natural ways of learning.” (p. 49.)
“Most teachers are satisfactory.” (p. 50.)
In reference to students who study mathematics in order
to “help them understand some other topic, for instance,
engineering:
“In such a case, the need is more for training in procedures and
algorithms rather than in the proper understanding of
mathematics itself.” (p. 51.)
When I joined the faculty of Purdue University in 1987
and began the development of new ways of teaching
Calculus (in courses largely populated by engineering
students) I found the above opinion to be the prevailing
view in the mathematics department. I was told that this
was the position of the Engineering and Science Faculty
and therefore the department’s calculus courses
emphasized training in procedures and algorithms. I
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ZDM 2001 Vol. 33 (3)
decided to check this before developing alternative
strategies. Together with my colleague, Keith
Schwingendorf, we contacted the leadership of every
Engineering and Science department (and all of them can
be found at Purdue) and asked for the names of faculty in
their department who could give us definitive information
about what their department wanted students to bring
them from their calculus courses. In some cases the chair
responded, in others an individual faculty member was
designated and in others, we met with a committee. To all
we said, “You require your students to take courses in
calculus. What would you like them to learn in these
courses?”
The responses, which did not vary very much were
astounding. The immediate response was in every case
something like: “Well certainly don’t emphasize
procedures and algorithms. We can quickly teach them to
do that with Maple or Mathematica. And don’t bother
much with applications. We prefer to teach the X (here
read one or another branch of Engineering or Science)
material ourselves because we are the experts. We want
you to teach them the mathematics in which you are the
experts.” The implication, occasionally explicit, was a
concern that in general, mathematicians might mess
things up in a field in which they might have had little
training.
The conversation then went on to consider
understanding mathematics and most of the engineering
and science faculty expressed the view that the proper
understanding of mathematics itself was what they hoped
their students would get from their mathematics courses.
One individual, who taught Physical Chemistry put it
rather succinctly. He said: “All I want is that if I am
teaching students who have studied calculus and want to
explain some physical concept, like work, in terms of
adding up a large number of small approximations by
constants and passing to a limit, then I would like my
students to act as if this general procedure was something
they had seen before.”
At the time my conclusion was, and it has remained so,
that this emphasis on procedures for engineering and
science students, advocated in the above statement by
Baumslag, is an idea that comes not from Engineering
and Science faculty, but from those who teach
mathematics! This is not the place to discuss why this had
ocurred, but I do wish that Baumslag had.
I will not suggest it represents an opinion, but I wish
someone had pointed out to the author that if the audience
for this book includes the United States, then a number of
potential readers will be offended by the use of the term,
“denigrating” (p. 52.)
I do not disagree with all of Baumslag’s viewpoints.
For example, I applaud his preference for texts that show
how results are obtained as opposed to those that “…use
a very terse style which require one to begin at the
beginning and master each step perfectly before
proceeding…” that “…assemble the precise material
needed, and faultlessly and without explaining any
connections, derive the major results in the most general
form.” (p. 73). But I find far more questionable opinions
than those I can support.
The strength of this book is that it touches on so much
ZDM 2001 Vol. 33 (3)
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of what one should be thinking about with respect to
undergraduate mathematics education. If this material
were given a better treatment, I would recommend that
beginning college teachers read it. But the overemphasis
of trivia, banal suggestions, and unfortunate opinions
presented with no regard to supporting evidence leads me
to conclude that the experienced teacher will learn little
from it and the novice can probably find better
introductions to teaching mathematics in an essentially
traditional manner, such as the recent book by Krantz on
the same topic.
____________
Author
Dubinsky, Ed, Prof. Dr., 265 North Woods Rd., Hermon, NY
13652, USA (May – Oct.); 5638 Springdale Rd., Apt. 3,
Cincinnati, OH 45251, USA (Oct. – May).
E-mail: edd@mcs.kent.edu
Internet: http://trident.mcs.kent.edu/~edd/
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